# Causal order in quantum coordinates

Updated: Apr 14

In today’s blog we talk to Dr. Lucien Hardy of the Perimeter Institute, PhD from Durham University, author of Hardy’s Theorem on non-locality, on his operational framework with indefinite causal structure in developing a quantum gravity theory.

**David**: You seem to have generalized the equivalence principle. What about it is quantum? Is there a sense in which one recovers the general relativistic equivalence principle within the quantum equivalence principle?

**Lucien**: The thing that is quantum about it particularly is that it uses a quantum coordinate system instead of a classical coordinate system. Actually, you do. In some sense you can think of it as a two-step process. First of all you find a quantum coordinate system in which the causal structure is definite. And then within that its almost trivial to make a further transformation so that you have local inertial behavior. So in some sense the quantum part of it is the thing that takes you to definite causal structure. And then there’s a further classical transformation which is just Einstein’s transformation that takes you into a local inertial coordinate system.

**David**: So there’s no such thing as a global coordinate system in which causality is definite?

**Lucien**: That’s exactly right. That’s analogous to the idea that there’s no such thing as a global coordinate system where you have inertial behavior everywhere. There’s no such global quantum coordinate system where you have global definite causal structure in general.

**David**: How did you come to this idea?

**Lucien**: I’d been bothered for a long time by indefinite causal structures. In 2005, I wrote a paper on this. It seemed to me that this was the crux of the problem of quantum gravity. People have different starting points. For me this seemed like the starting point. If you bring quantum theory and general relativity together, what you get is indefinite causal structure, and this seems like the real place where there’s a conceptual clash between the two theories. For a long time I was worried about that and in particular if you have indefinite causal structure then how do you impose the usual sorts of features that we have in physical theories like causality? It’s difficult to know what causality means if you don’t have that causal structure. I’ve been aware of this quantum reference frame discussion for a long time. But then Flaminia Giacomini, Esteban Castro-Ruiz, and Časlav Brukner, were able to put the reference frame idea in a different sort of format, which was much more appealing to me. And then I was able to see that actually you could do something like a quantum reference frame transformation that would get rid of indefinite causal structure locally. Actually, the idea that they introduced didn’t quite work for my purposes so I had to redo that more in the spirit of general relativity.

**David**: You seem to be treating the ideas of GR as fundamental. Some would say that GR is an emergent theory. How do you make sense of the thermodynamic aspect of GR in this respect?

**Lucien**: Since we don’t have a theory of quantum gravity, several options are on the table. I wouldn’t say that general relativity is fundamental as such. But rather, it provides a greater example of how you can go about unifying two theories. What Einstein did was to unify Newtonian gravity on the one hand, with special relativistic field theory on the other hand. Those two theories didn’t fit together and the impossibility proofs in the early part of the 20th century showed that you couldn’t put gravity into a special relativistic field theory. So Einstein found a way to unify them. And the equivalence principle was right at the heart of that unification. He showed that even though you couldn’t do it globally, you could patch together a locally classical Newtonian gravity picture where the equivalence principle holds into a field theory. From a construction point of view, if you’re interested in building theories, that’s a very interesting idea. Perhaps it can be used again. So that’s what I’m aiming for. That being said, I have to say I do think there’s something fundamental about general relativity that hasn’t been fully taken on board. And I think people are trying to avoid it. Our natural disposition is to go back to this idea that there is this universal time that evolves in the background forwards. And general relativity completely destroys that idea. Well, that’s not quite true. In general relativity you can recover that idea if you impose this sort of canonical formulation on GR.

**David**: Where you separate space and time?

**Lucien**: Yeah, where you separate space and time and have this evolution. And, it seems to me that’s just the wrong thing to do. It goes very much against the spirit of general relativity even though it seems to work. In terms of the derivation of GR in terms of a thermodynamic picture, I haven’t studied those approaches in enough detail to know if they have that universal background time point of view. But, I think they do. And so I would criticize them on that basis. It’s interesting that it’s successful but I suspect it’s the wrong approach.

**David**: So you might say that the thermodynamic likeness is a sort of coincidence?

**Lucien**: Yeah. There’s bound to be something happening there but it’s probably a coincidence. One thing I’ve worked on is formulating general relativity in a probabilistic fashion. And that’s quite difficult because generally when you think about probabilistic formulations of a theory, what you do is to take some initial distribution in time and you evolve it forward in time. But even in classical general relativity, if you have probabilistic ignorance, that ignorance is likely to entail the causal structure. So your ignorance wouldn’t necessarily be of the type that it pertained to an initial time. You could have ignorance over different causal structures that couldn’t be understood in that way. So I think that bringing probabilities together with general relativity is an interesting endeavor. It’s sort of a warm-up exercise for quantum gravity.

**David**: So is it fair to say that causality like inertial frames is an approximation? Like spacetime is globally curved, causality is globally indefinite?

**Lucien**: I guess it would be an approximation in the same way that having a local inertial system is an approximation. There’s always a sense in which it isn’t quite true. It’s true to first order. But there’s always these second order corrections. So I think the same would be true in quantum gravity. I haven’t yet written down the relevant equations. That’s sort of the next project.

**David**: In a sense it’s fair to say that the global nature of causality is indefinite.

**Lucien**: Yeah. That’s right.

**David**: And that’s a very radical statement.

**Lucien**: I think it is radical. Yes. But I also think it is more or less inevitable if you take seriously this conceptual clash between quantum theory and general relativity. And I don’t think people are taking this conceptual clash sufficiently seriously. They’re trying to reimpose ideas that take you back to definite causal structure. If you take a sort of graviton type of approach, a particle physics approach to quantum gravity, then you have a background definite causal structure and gravitons are just small perturbations on that background which admits some indefinite causal structure but that is completely tamed in the wrong way. Or in the canonical approaches to quantum gravity like loop quantum gravity, certainly the ones that are based on Hamiltonian physics, what they do is exactly what I was talking about earlier. You do a 3+1 splitting into space and time and then you evolve the state forward. This just doesn’t take seriously the indefinite causal structure. There are other approaches in that community which are perhaps better suited to the problem of indefinite causal structure.

**David**: Do you have a sense of how your approach will solve the measurement problem?

**Lucien**: It’s a good question. It’s something I worry about. So I have a broad philosophy here and then some more detailed ideas. The broad philosophy is this. In the 1990’s I went to many conferences where people talked about the measurement problem and different interpretations of quantum theory with different solutions. I went to these conferences year after year and again and again it was the same ideas, the same solutions. I didn’t see any root progress along those lines. I think the problem is similar to the problem that was faced by Newton and his contemporaries in his theory of gravity. It was non-local. There’s also no mediating substance. There’s nothing mediated the force between two bodies at a distance. It is instantaneous and they didn’t like that. Now you could have spent a lot of time trying to resolve that problem. There could have been many conferences devoted to solving this problem. But it’s unlikely that if you attack that problem straight on, you would have gotten to the correct solution. The correct solution was to move beyond Newtonian gravity to general relativity and then suddenly you had a theory which is local, there is a mediating substance, the metric. And so the correct approach in that case was to construct the new theory and then the solution becomes evident. Actually, even in retrospect it is academically interesting that in general relativity you can take the limit as c tends to infinity, and you get this Newton-Cartan version of general relativity in which the metric splits up into a space part and a time part and the gravitational force is mediated by these parts of the metric. It’s instantaneous but there’s a mediating substance. But that sort of understanding just wasn’t conceivable back then. But I think we’re in the same situation now. We’re trying to solve a problem in terms of just quantum theory. But we know there’s this big revolution coming where we get a theory of quantum gravity, and it’s likely to be quite a different looking theory from quantum theory. So that’s the general philosophy. For the particular terms, one thing I find very promising is when you go to this idea of having quantum coordinate systems, you can also construct quantum diffeomorphisms. In classical general relativity, the beables – the things that are real – are properties that are invariant under coordinate transformations or abstractly general diffeomorphisms. It’s a very strange theory actually. You might say that what’s happening here on this part of the manifold is real. But you can hit the solution with a diffeomorphism or a general coordinate transformation, and move everything to a different part of the manifold. You haven’t changed the physics. What’s really real in general relativity are relational properties that survive after you hit it with a general coordinate transformation. Now, in the context of quantum gravity, the natural thing to say is that what’s real is what survives after a quantum diffeomorphism. And this is a different beast from a classical diffeomorphism. It’s a more general thing. We don’t have good intuition as to what it does. My hope is that if you understand that properly, you might see a solution to the measurement problem. But I don’t have any constructive ideas along those lines yet. That’s just a hope.

**David**: Do you have a sense of whether this mathematical structure you’re putting together is simple, natural, or in some sense similar to the simplicity or naturalness of general coordinate transformations of GR?

**Lucien**: It’s a good question. I hope it will be. But I’m not seeing that yet. The work I did on classical general relativity, trying to make it probabilistic that I mentioned, its framework was not very natural looking. It didn’t have that sort of natural feeling. And so that concerns me. The idea of a quantum reference frame and the idea of transforming to local definite causal structure is a nice one. So I’m hopeful that mathematically it will work out. One of the problems, though, is that the project suggests that you should get all the objects that you get in classical general relativity. So we have an analog to the equivalence principle, the quantum equivalence principle. We have an analog to general coordinate systems, which are quantum coordinate systems, and transformations. We have that analog. It would be nice to have an analog to the manifold structure. And I speculated in the paper on the archive about what the analog would be. But it seems that this quantum manifold structure is a much more complicated thing because in addition to having four dimensions associated with the usual manifold structure of general relativity, there is a sort of extra direction to do with all the possibilities you could have. And that extra dimension is infinite dimensional, which doesn’t bode well for mathematical simplicity. So it’s a good question and one I’m concerned about but I don’t know how it’s going to pan out.

**David**: Is it fair to say that you are discovering that causality in principle is much less fundamental than we thought?

**Lucien**: No. I wouldn’t say that actually. I think that causality is still there if you view physics in the right way. Would one say that inertial behavior was not fundamental in general relativity? In one sense it is very fundamental because the fact that you can always transform to an inertial reference frame plays a central role in our understanding of the theory. If you are falling in space you feel that very directly. So the same thing would be true in quantum gravity. Causality would still very much be there but it would be a quantum frame dependent thing.

**David**: We think of physics as the subject dealing with matter moving in space as time goes by. We are now talking about superpositions not only of matter, space, but of causality. What is physics about?

**Lucien**: It is true. In that view of there being a sort of global, more or less fixed spacetime, and matter just evolves in that, definitely goes against what I am working on. But still, locally, within that picture you can find local quantum coordinate systems where you do see this behavior. Is that fundamental? Well, if it plays a deep role in constructing the theory of quantum gravity then it is fundamental. But it’s not the same as this previous global picture that you alluded to.

**David**: From an outsider’s perspective you seem to have generated lots of interest. What are the reactions?

**Lucien**: I haven’t actually got a strong sense of that. One thing that’s been fun is that there’s this John Templeton funded network called QISS, Quantum Information Structure of Spacetime, organized by Carlo Rovelli. And there are places around the world that are taking part in it. And it brings people with a quantum foundations type of background, like myself, together with people from a quantum gravity background like Carlo. Caslav Brukner and collaborators, including Flaminia Giacomini who is now at Perimeter Institute but who was a student of his, have also been thinking about the quantum equivalence principle. So it’s starting to become something that people are thinking about. But you really have to go to those people to get a sense of that.

**David**: Do you have a sense of how long or deep one has to dive into a new idea before one recognizes if there’s something to it? For example, string theory has been around for a long time. It seems to be getting quieter with time. Has it stalled? What about LQG? Do these things experience a peak and then interest drops?

**Lucien**: Yeah. I have the same sense. Those communities are still very active in various ways but those approaches seem to be quieter, especially if construed as approaches to quantum gravity. In particular, string theory, you don’t hear a lot on that front. I haven’t invested the time to understand those approaches properly so I couldn’t comment on the technical problems. I do think, in general, it’s good to have a variety of different approaches. One of the problems with the way physics is structured is that it tends to favor everyone working on the same approach which can be good for your career. That’s a drawback. Anything which is different from those approaches I think is a good idea. In terms of this approach, time will tell whether it’s a good idea or not.

**David**: Do you think the most exciting things to come from this approach are still in the future?

**Lucien**: Yeah. I think the group of people in quantum foundations, quantum information, are relatively new to quantum gravity. So there’s a steep learning curve for us.

**David**: Was it difficult to put together this notion of quantum coordinate systems?

**Lucien**: It took me a long time to work out the right idea, or what I think is the right idea. First of all, I needed to reconstrue quantum theory in such a light that one could have this idea. And to do that I went to the path integral formulation. The reason for that is that in standard quantum theory you have a state in time and you evolve it forward. If that state has definite position, for example, it doesn’t have definite momentum, so a quantum state at a given time cannot give you a full classical description. In the path integral formulation, it’s kind of weird that each path is actually a full classical description. It’s not one that you can prepare but it is one that you can speak about in the formalism. So the objects that enter into the path integral are these full classical descriptions. And if you want to talk about four dimensional coordinate systems and transformations in those diffeomorphisms, then you need to talk about classically described objects. So that was the hard thing for me, to get to that point. But the path integral approach wasn’t quite the right framework so I needed to reconstrue it in a way that was more suitable to this quantum reference frame point of view. Having the idea that there should be a quantum equivalence principle, it took me several months to make it work. The paper that I’m almost finished on at the moment is on symmetry. And this is also a big part of the problem because there is a time asymmetric causality condition. The causality condition is that choices made in the future can’t influence the past but not vice versa. Choices made in the past do influence the future. And then the causality condition is just a way of mathematizing that statement. The causality condition in standard operational frameworks is that the deterministic effect is unique. That means that there is only one way of ignoring the future. If there were two different ways of ignoring the future, you could choose between them. Then whichever choice you made would have an influence on the properties in the past in these sorts of frameworks. So I’ve been working on making this whole thing time symmetric. So now you have a causality principle that is time symmetric. And this is important for this project of the quantum equivalence principle because the thing I want to do with that principle is to transform to a frame of reference where you have definite causal structure. And it would be nice if that were a time symmetric notion of causal structure. So this has been a tremendous problem to solve that has taken me several months of hard work.

**David**: From what perspective would this time symmetry be nice to have? Just based on this being a feature of the laws of physics generally?

**Lucien**: Well, for one, general relativity is time symmetric. So if one is trying to pursue this analogy with general relativity, it would be nice if the causality principle that you’re transforming was time symmetric. And the other is that time asymmetry seems to tie in more tightly to a background frame of reference, to a definite background. And it would be nice to be away from that. I guess I have a strong intuition that time asymmetry is not the right thing.

**David**: Much of your intuition is based on assuming this parallel with GR. I imagine people being critical of this. They might say that you need a fundamental principle or symmetry behind constructing a more general theory. Have you experienced that?

**Lucien**: I think everyone is using intuition initially and then you work on it and get something more concrete. In this case I now have a very concrete time symmetric notion of causality that applies to this way of thinking. So I think that’s what people do generally.

**David**: One last thing. I want to focus a little on the sociology of quantum gravity studies. One would imagine that if you see better ideas somewhere else, one would migrate toward those ideas. Do you see people moving away from one approach and towards another?

**Lucien**: I think these things happen much more slowly than one might think. Of course, I have less experience of the quantum gravity community than I do of the quantum foundations community. The quantum foundations community has moved. A large section of that community is now thinking about physics in operational ways using general property theories, but also employing ideas from quantum information. In quantum gravity I’m less equipped to say. From talking to people in that community, I get the sense that they are very open minded. At Perimeter we are very lucky that we have representatives of all these different communities and I have good conversations with people from string theory, from loop quantum gravity, etc. There’s a difference between the discussions you have over coffee or, nowadays over zoom meetings, and people actually changing their research direction. But that just takes a longer period of time. I mean the string community has been of now very interested in quantum information. So that’s been a big shift in that community compared with what they were doing ten years ago. That’s certainly to be welcomed.

**David**: Thank you for doing this.

**Lucien**: You’re very welcome.

**David**: I’ll transcribe it and send it to you.

**Lucien**: Ok. So you’ve been doing this blog for a while now?

**David**: No. I just started. This is my way of exploring other topics that I find interesting.

**Lucien**: It’s a great idea.

**David**: Thank you Professor!